| $z$ | $\mathrm{P}(0 \leq Z \leq z)$ |
| 0.6 | 0.226 |
| 0.8 | 0.288 |
| 1.0 | 0.341 |
| 1.2 | 0.385 |
| 1.4 | 0.419 |
For a positive number $t$, the random variable $X$ follows a normal distribution $\mathrm{N}(1, t^2)$.
$$\mathrm{P}(X \leq 5t) \geq \frac{1}{2}$$
For all positive numbers $t$ satisfying this condition, find the maximum value of $\mathrm{P}(t^2 - t + 1 \leq X \leq t^2 + t + 1)$ using the standard normal distribution table below, and let this value be $k$. Find the value of $1000 \times k$. [4 points]
\begin{center}
\begin{tabular}{ | c | c | }
\hline
$z$ & $\mathrm{P}(0 \leq Z \leq z)$ \\
\hline
0.6 & 0.226 \\
\hline
0.8 & 0.288 \\
\hline
1.0 & 0.341 \\
\hline
1.2 & 0.385 \\
\hline
1.4 & 0.419 \\
\hline
\end{tabular}
\end{center}